OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 OptimalMöbiusTransformation forInformationVisualizationandMeshing MarshallBern XeroxPaloAltoResearchCtr. DavidEppstein Univ.ofCalifornia,Irvine Dept.ofInformationandComputerScience
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 WhatareMöbiustransformations? Fractionallineartransformationsofcomplexnumbers: z → ( az + b )/( cz + d ) Butwhatdoesitmeangeometrically? Howtogeneralizetohigherdimensions? Whatisitgoodfor?
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 WhatareMöbiustransformations?(II) Circle-preservingmapsfromtheplanetoitself Moreintuitive Generalizesnicelytospheres,higherdimensionalspaces Notveryconcrete
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 WhatareMöbiustransformations(III) Inversion:mapradiiofcircletosameray sothatproductofdistancesfromcenter=radius 2 Möbiustransformation=compositionofmultipleinversions Moreconcrete,stillgeneralizesnicely
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 WhatareMöbiustransformations?(IV) Viewplaneasboundaryofhalfspacemodelofhyperbolicspace Möbiustransformationsofplane ↔ hyperbolicisometries Esoteric Mostusefulforouralgorithms
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 OptimalMöbiustransformation: Givenaplanar(orhigherdimensional)inputconfjguration SelectaMöbiustransformation fromthe(six-dimensionalorhigher)spaceofallMöbiustransformations Thatoptimizestheshapeofthetransformedinput Typicallymin-maxormax-minproblems: maximizemin(setoffunctionsdescribingtransformedshapequality)
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Application:sphericalgraphdrawing Theorem[Koebe,Thurston]:Anyplanargraphcanberepresented bydisjointdisksonaspheresotwoverticesadjacentifftwodiskstangent Formaximalplanargraphs,uniqueuptoMöbiustransformation Othergraphscanbemademaximalplanarbyaddingvertexineachface Optimizationproblem: Finddiskrepresentationof G maximizingminimumdiskradius or,givenonediskrepresentation,fjndMöbiustransformationmaximizingminradius Solutionalsoturnsouttodisplayallsymmetriesofinitialembeddedgraph
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Application:conformalmeshgeneration Givensimply-connectedplanardomaintobemeshed Maptosquare,useregularmesh,invertmaptogivemeshinoriginaldomain Differentpointsofdomainmayhavedifferentrequirementsforelementsize Tominimize#elements,mapregionsrequiringsmallsizetolargeareasofsquare ConformalmapisuniqueuptoMöbiustransformation OptimizationProblem: Findconformalmapmaximizingmin(sizerequirement*localexpansionfactor) tominimizeoverallnumberofelementsproduced
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Application:hyperbolicbrowser [Lamping,Rao,andPirolli,1995] Techniqueforviewinglargewebsitesorotherstructuredinformation bylayingoutinformationinhyperbolicspace Allows“fjsheyeview”:close-uplookatdetailsofsomepointinsite globalstructureofsitevisibletowardsboundaryofhyperbolicmodel Thefartherawayapointisfromtheviewpoint(inhyperbolicdistance) thesmallertheinformationitrepresentswillbedisplayed Optimizationproblem: Findgoodinitialviewpointforhyperbolicbrowser inordertomakeoverallsiteasvisibleaspossible Maximizeminimumsizeofdisplayedobject or Maximizeminimumseparationbetweentwoobjects
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Application:brainfmatmapping [Hurdaletal.1999] Problem:visualizethehumanbrain Alltheinterestingstuffisonthesurface Butdiffjcultbecausethesurfacehascomplicatedfolds Approach:fjndquasi-conformalmappingbrain → plane Thencanvisualizebrainfunctionalunitsasregionsofmappedplane Avoidsdistortinganglesbutareascanbegreatlydistorted Asinmeshgen.problem,mappinguniqueuptoMöbiustransformation Optimizationproblem: Givenmap3dtriangulatedsurface → plane, fjndMöbiustransformationminimizingmax(areadistortionoftriangle)
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 OptimalMöbiusAlgorithm Keycomponents: QuasiconvexProgramming HyperbolicGeometry
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Quasiconvexfunctions Levelsetsarenestedconvexcurves Innercurvescorrespondtosmallerfunctionvalues (Liketopographicmapofopenpitmine)
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Quasiconvexprogramming [Amenta,Bern,Eppstein1999] Given n quasiconvexfunctions f i max( f i ( x ))isalsoquasiconvex problemissimplytocompute x minimizingmax( f i ( x )) CanbesolvedexactlywithO( n )constant-sizesubproblems usinglow-dimensionallinear-programming-typetechniques Canbesolvednumericallybyhill-climbingorotherlocaloptimizationmethods
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Hyperbolicspace(Poincarémodel) Interiorofunitsphere;linesandplanesaresphericalpatchesperpendiculartounitsphere
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Twomodelsofhyperbolicspace KleinModel Hyperbolicobjectsarestraightorconvex ifftheirmodelisstraightorconvex Anglesareseverelydistorted Hyperbolicsymmetriesaremodeledas Euclideanprojectivetransformations PoincaréModel Anglesinhyperbolicspace equalEuclideananglesoftheirmodels Straightness/convexitydistorted Hyperbolicsymmetriesaremodeledas Möbiustransformations
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Möbiustransformationandhyperbolicgeometry Quasiconvexprogrammingworksequallywellinhyperbolicspace Duetoconvexity-preservingpropertiesofKleinmodel ButspaceofMöbiustransformationsisnothyperbolicspace... ViewMöbiustransformationaschoiceofPoincarémodel Factortransformationsinto choiceofcenterpointinhyperbolicmodel(affectsshape) Euclideanrotationaroundcenterpoint(doesn’taffectshape)
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 OptimalMöbiustransformationalgorithm Representoptimizationproblemobjectivefunction asmaxofsetofquasiconvexfunctions wherefunctionargumentishyperboliccenterpointlocation Hardpart:provingthatourobjectivefunctionsarequasiconvex Solvequasiconvexprogram UsecenterpointlocationtofjndMöbiustransformation
OptimalMöbiusTransformation D.Eppstein,UCIrvine,WADS2001 Conclusions FormulateseveralinterestingapplicationsasMöbiusoptimization CansolveviaLP-typetechniquesorhill-climbing Interestinguseofhyperbolicmethodsincomputationalgeometry but... Detailsofexactalgorithmmaybediffjculttoimplement (seeGärtnerforsimilardiffjcultiesinLP-typemin-volumeellipsoid) Notabletoprovequasiconvexityinsomecases e.g.givennumberx,triangleTatinfjnityinhyperbolic3-space arepointsfromwhichTsubtendssolidangle>xconvex?
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